Integrand size = 19, antiderivative size = 207 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {256 d^4 \sqrt {a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac {512 b d^4 \sqrt {a+b x}}{21 (b c-a d)^6 \sqrt {c+d x}} \]
-2/7/(-a*d+b*c)/(b*x+a)^(7/2)/(d*x+c)^(3/2)+4/7*d/(-a*d+b*c)^2/(b*x+a)^(5/ 2)/(d*x+c)^(3/2)-32/21*d^2/(-a*d+b*c)^3/(b*x+a)^(3/2)/(d*x+c)^(3/2)+64/7*d ^3/(-a*d+b*c)^4/(d*x+c)^(3/2)/(b*x+a)^(1/2)+256/21*d^4*(b*x+a)^(1/2)/(-a*d +b*c)^5/(d*x+c)^(3/2)+512/21*b*d^4*(b*x+a)^(1/2)/(-a*d+b*c)^6/(d*x+c)^(1/2 )
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\frac {2 \left (-7 a^5 d^5+35 a^4 b d^4 (3 c+2 d x)+70 a^3 b^2 d^3 \left (3 c^2+12 c d x+8 d^2 x^2\right )+70 a^2 b^3 d^2 \left (-c^3+6 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )+7 a b^4 d \left (3 c^4-8 c^3 d x+48 c^2 d^2 x^2+192 c d^3 x^3+128 d^4 x^4\right )+b^5 \left (-3 c^5+6 c^4 d x-16 c^3 d^2 x^2+96 c^2 d^3 x^3+384 c d^4 x^4+256 d^5 x^5\right )\right )}{21 (b c-a d)^6 (a+b x)^{7/2} (c+d x)^{3/2}} \]
(2*(-7*a^5*d^5 + 35*a^4*b*d^4*(3*c + 2*d*x) + 70*a^3*b^2*d^3*(3*c^2 + 12*c *d*x + 8*d^2*x^2) + 70*a^2*b^3*d^2*(-c^3 + 6*c^2*d*x + 24*c*d^2*x^2 + 16*d ^3*x^3) + 7*a*b^4*d*(3*c^4 - 8*c^3*d*x + 48*c^2*d^2*x^2 + 192*c*d^3*x^3 + 128*d^4*x^4) + b^5*(-3*c^5 + 6*c^4*d*x - 16*c^3*d^2*x^2 + 96*c^2*d^3*x^3 + 384*c*d^4*x^4 + 256*d^5*x^5)))/(21*(b*c - a*d)^6*(a + b*x)^(7/2)*(c + d*x )^(3/2))
Time = 0.25 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {55, 55, 55, 55, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \int \frac {1}{(a+b x)^{7/2} (c+d x)^{5/2}}dx}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}}dx}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {2 d \int \frac {1}{(a+b x)^{3/2} (c+d x)^{5/2}}dx}{b c-a d}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {2 d \left (-\frac {4 d \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}}dx}{b c-a d}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {2 d \left (-\frac {4 d \left (\frac {2 b \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 (b c-a d)}+\frac {2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {2 d \left (-\frac {4 d \left (\frac {4 b \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^2}+\frac {2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2}{\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}\) |
-2/(7*(b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2)) - (10*d*(-2/(5*(b*c - a *d)*(a + b*x)^(5/2)*(c + d*x)^(3/2)) - (8*d*(-2/(3*(b*c - a*d)*(a + b*x)^( 3/2)*(c + d*x)^(3/2)) - (2*d*(-2/((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2 )) - (4*d*((2*Sqrt[a + b*x])/(3*(b*c - a*d)*(c + d*x)^(3/2)) + (4*b*Sqrt[a + b*x])/(3*(b*c - a*d)^2*Sqrt[c + d*x])))/(b*c - a*d)))/(b*c - a*d)))/(5* (b*c - a*d))))/(7*(b*c - a*d))
3.16.21.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Time = 0.53 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(-\frac {2}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {10 d \left (-\frac {2}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {8 d \left (-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 d \left (-\frac {2}{\left (-a d +b c \right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}-\frac {4 d \left (-\frac {2 \sqrt {b x +a}}{3 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 \left (a d -b c \right )^{2} \sqrt {d x +c}}\right )}{-a d +b c}\right )}{-a d +b c}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\) | \(215\) |
| gosper | \(-\frac {2 \left (-256 x^{5} b^{5} d^{5}-896 x^{4} a \,b^{4} d^{5}-384 x^{4} b^{5} c \,d^{4}-1120 x^{3} a^{2} b^{3} d^{5}-1344 x^{3} a \,b^{4} c \,d^{4}-96 x^{3} b^{5} c^{2} d^{3}-560 x^{2} a^{3} b^{2} d^{5}-1680 x^{2} a^{2} b^{3} c \,d^{4}-336 x^{2} a \,b^{4} c^{2} d^{3}+16 x^{2} b^{5} c^{3} d^{2}-70 x \,a^{4} b \,d^{5}-840 x \,a^{3} b^{2} c \,d^{4}-420 x \,a^{2} b^{3} c^{2} d^{3}+56 x a \,b^{4} c^{3} d^{2}-6 x \,b^{5} c^{4} d +7 a^{5} d^{5}-105 a^{4} b c \,d^{4}-210 a^{3} b^{2} c^{2} d^{3}+70 a^{2} b^{3} c^{3} d^{2}-21 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right )}{21 \left (b x +a \right )^{\frac {7}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}\) | \(356\) |
-2/7/(-a*d+b*c)/(b*x+a)^(7/2)/(d*x+c)^(3/2)-10/7*d/(-a*d+b*c)*(-2/5/(-a*d+ b*c)/(b*x+a)^(5/2)/(d*x+c)^(3/2)-8/5*d/(-a*d+b*c)*(-2/3/(-a*d+b*c)/(b*x+a) ^(3/2)/(d*x+c)^(3/2)-2*d/(-a*d+b*c)*(-2/(-a*d+b*c)/(d*x+c)^(3/2)/(b*x+a)^( 1/2)-4*d/(-a*d+b*c)*(-2/3/(a*d-b*c)/(d*x+c)^(3/2)*(b*x+a)^(1/2)+4/3*b/(a*d -b*c)^2*(b*x+a)^(1/2)/(d*x+c)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 999 vs. \(2 (171) = 342\).
Time = 4.12 (sec) , antiderivative size = 999, normalized size of antiderivative = 4.83 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx =\text {Too large to display} \]
2/21*(256*b^5*d^5*x^5 - 3*b^5*c^5 + 21*a*b^4*c^4*d - 70*a^2*b^3*c^3*d^2 + 210*a^3*b^2*c^2*d^3 + 105*a^4*b*c*d^4 - 7*a^5*d^5 + 128*(3*b^5*c*d^4 + 7*a *b^4*d^5)*x^4 + 32*(3*b^5*c^2*d^3 + 42*a*b^4*c*d^4 + 35*a^2*b^3*d^5)*x^3 - 16*(b^5*c^3*d^2 - 21*a*b^4*c^2*d^3 - 105*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)* x^2 + 2*(3*b^5*c^4*d - 28*a*b^4*c^3*d^2 + 210*a^2*b^3*c^2*d^3 + 420*a^3*b^ 2*c*d^4 + 35*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*b^6*c^8 - 6*a^ 5*b^5*c^7*d + 15*a^6*b^4*c^6*d^2 - 20*a^7*b^3*c^5*d^3 + 15*a^8*b^2*c^4*d^4 - 6*a^9*b*c^3*d^5 + a^10*c^2*d^6 + (b^10*c^6*d^2 - 6*a*b^9*c^5*d^3 + 15*a ^2*b^8*c^4*d^4 - 20*a^3*b^7*c^3*d^5 + 15*a^4*b^6*c^2*d^6 - 6*a^5*b^5*c*d^7 + a^6*b^4*d^8)*x^6 + 2*(b^10*c^7*d - 4*a*b^9*c^6*d^2 + 3*a^2*b^8*c^5*d^3 + 10*a^3*b^7*c^4*d^4 - 25*a^4*b^6*c^3*d^5 + 24*a^5*b^5*c^2*d^6 - 11*a^6*b^ 4*c*d^7 + 2*a^7*b^3*d^8)*x^5 + (b^10*c^8 + 2*a*b^9*c^7*d - 27*a^2*b^8*c^6* d^2 + 64*a^3*b^7*c^5*d^3 - 55*a^4*b^6*c^4*d^4 - 6*a^5*b^5*c^3*d^5 + 43*a^6 *b^4*c^2*d^6 - 28*a^7*b^3*c*d^7 + 6*a^8*b^2*d^8)*x^4 + 4*(a*b^9*c^8 - 3*a^ 2*b^8*c^7*d - 2*a^3*b^7*c^6*d^2 + 19*a^4*b^6*c^5*d^3 - 30*a^5*b^5*c^4*d^4 + 19*a^6*b^4*c^3*d^5 - 2*a^7*b^3*c^2*d^6 - 3*a^8*b^2*c*d^7 + a^9*b*d^8)*x^ 3 + (6*a^2*b^8*c^8 - 28*a^3*b^7*c^7*d + 43*a^4*b^6*c^6*d^2 - 6*a^5*b^5*c^5 *d^3 - 55*a^6*b^4*c^4*d^4 + 64*a^7*b^3*c^3*d^5 - 27*a^8*b^2*c^2*d^6 + 2*a^ 9*b*c*d^7 + a^10*d^8)*x^2 + 2*(2*a^3*b^7*c^8 - 11*a^4*b^6*c^7*d + 24*a^5*b ^5*c^6*d^2 - 25*a^6*b^4*c^5*d^3 + 10*a^7*b^3*c^4*d^4 + 3*a^8*b^2*c^3*d^...
\[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {9}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1964 vs. \(2 (171) = 342\).
Time = 1.06 (sec) , antiderivative size = 1964, normalized size of antiderivative = 9.49 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
2/3*sqrt(b*x + a)*(14*(b^9*c^5*d^6*abs(b) - 5*a*b^8*c^4*d^7*abs(b) + 10*a^ 2*b^7*c^3*d^8*abs(b) - 10*a^3*b^6*c^2*d^9*abs(b) + 5*a^4*b^5*c*d^10*abs(b) - a^5*b^4*d^11*abs(b))*(b*x + a)/(b^13*c^11*d - 11*a*b^12*c^10*d^2 + 55*a ^2*b^11*c^9*d^3 - 165*a^3*b^10*c^8*d^4 + 330*a^4*b^9*c^7*d^5 - 462*a^5*b^8 *c^6*d^6 + 462*a^6*b^7*c^5*d^7 - 330*a^7*b^6*c^4*d^8 + 165*a^8*b^5*c^3*d^9 - 55*a^9*b^4*c^2*d^10 + 11*a^10*b^3*c*d^11 - a^11*b^2*d^12) + 15*(b^10*c^ 6*d^5*abs(b) - 6*a*b^9*c^5*d^6*abs(b) + 15*a^2*b^8*c^4*d^7*abs(b) - 20*a^3 *b^7*c^3*d^8*abs(b) + 15*a^4*b^6*c^2*d^9*abs(b) - 6*a^5*b^5*c*d^10*abs(b) + a^6*b^4*d^11*abs(b))/(b^13*c^11*d - 11*a*b^12*c^10*d^2 + 55*a^2*b^11*c^9 *d^3 - 165*a^3*b^10*c^8*d^4 + 330*a^4*b^9*c^7*d^5 - 462*a^5*b^8*c^6*d^6 + 462*a^6*b^7*c^5*d^7 - 330*a^7*b^6*c^4*d^8 + 165*a^8*b^5*c^3*d^9 - 55*a^9*b ^4*c^2*d^10 + 11*a^10*b^3*c*d^11 - a^11*b^2*d^12))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 8/21*(79*sqrt(b*d)*b^15*c^6*d^3 - 474*sqrt(b*d)*a*b^14*c^ 5*d^4 + 1185*sqrt(b*d)*a^2*b^13*c^4*d^5 - 1580*sqrt(b*d)*a^3*b^12*c^3*d^6 + 1185*sqrt(b*d)*a^4*b^11*c^2*d^7 - 474*sqrt(b*d)*a^5*b^10*c*d^8 + 79*sqrt (b*d)*a^6*b^9*d^9 - 511*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^13*c^5*d^3 + 2555*sqrt(b*d)*(sqrt(b*d)*sqrt(b* x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^12*c^4*d^4 - 5110*sqrt (b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^ 2*b^11*c^3*d^5 + 5110*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +...
Time = 1.78 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.31 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {32\,x^2\,\left (35\,a^3\,d^3+105\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{21\,b\,{\left (a\,d-b\,c\right )}^6}-\frac {14\,a^5\,d^5-210\,a^4\,b\,c\,d^4-420\,a^3\,b^2\,c^2\,d^3+140\,a^2\,b^3\,c^3\,d^2-42\,a\,b^4\,c^4\,d+6\,b^5\,c^5}{21\,b^3\,d^2\,{\left (a\,d-b\,c\right )}^6}+\frac {64\,d\,x^3\,\left (35\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {512\,b^2\,d^3\,x^5}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {256\,b\,d^2\,x^4\,\left (7\,a\,d+3\,b\,c\right )}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {x\,\left (140\,a^4\,b\,d^5+1680\,a^3\,b^2\,c\,d^4+840\,a^2\,b^3\,c^2\,d^3-112\,a\,b^4\,c^3\,d^2+12\,b^5\,c^4\,d\right )}{21\,b^3\,d^2\,{\left (a\,d-b\,c\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{b^2\,d^2}+\frac {x^4\,\left (3\,a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {a^3\,c^2\,\sqrt {a+b\,x}}{b^3\,d^2}+\frac {a\,x^2\,\sqrt {a+b\,x}\,\left (a^2\,d^2+6\,a\,b\,c\,d+3\,b^2\,c^2\right )}{b^3\,d^2}+\frac {a^2\,c\,x\,\left (2\,a\,d+3\,b\,c\right )\,\sqrt {a+b\,x}}{b^3\,d^2}} \]
((c + d*x)^(1/2)*((32*x^2*(35*a^3*d^3 - b^3*c^3 + 21*a*b^2*c^2*d + 105*a^2 *b*c*d^2))/(21*b*(a*d - b*c)^6) - (14*a^5*d^5 + 6*b^5*c^5 + 140*a^2*b^3*c^ 3*d^2 - 420*a^3*b^2*c^2*d^3 - 42*a*b^4*c^4*d - 210*a^4*b*c*d^4)/(21*b^3*d^ 2*(a*d - b*c)^6) + (64*d*x^3*(35*a^2*d^2 + 3*b^2*c^2 + 42*a*b*c*d))/(21*(a *d - b*c)^6) + (512*b^2*d^3*x^5)/(21*(a*d - b*c)^6) + (256*b*d^2*x^4*(7*a* d + 3*b*c))/(21*(a*d - b*c)^6) + (x*(140*a^4*b*d^5 + 12*b^5*c^4*d - 112*a* b^4*c^3*d^2 + 1680*a^3*b^2*c*d^4 + 840*a^2*b^3*c^2*d^3))/(21*b^3*d^2*(a*d - b*c)^6)))/(x^5*(a + b*x)^(1/2) + (x^3*(a + b*x)^(1/2)*(3*a^2*d^2 + b^2*c ^2 + 6*a*b*c*d))/(b^2*d^2) + (x^4*(3*a*d + 2*b*c)*(a + b*x)^(1/2))/(b*d) + (a^3*c^2*(a + b*x)^(1/2))/(b^3*d^2) + (a*x^2*(a + b*x)^(1/2)*(a^2*d^2 + 3 *b^2*c^2 + 6*a*b*c*d))/(b^3*d^2) + (a^2*c*x*(2*a*d + 3*b*c)*(a + b*x)^(1/2 ))/(b^3*d^2))
Time = 0.15 (sec) , antiderivative size = 1344, normalized size of antiderivative = 6.49 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx =\text {Too large to display} \]
int(1/(sqrt(c + d*x)*sqrt(a + b*x)*(a**4*c**2 + 2*a**4*c*d*x + a**4*d**2*x **2 + 4*a**3*b*c**2*x + 8*a**3*b*c*d*x**2 + 4*a**3*b*d**2*x**3 + 6*a**2*b* *2*c**2*x**2 + 12*a**2*b**2*c*d*x**3 + 6*a**2*b**2*d**2*x**4 + 4*a*b**3*c* *2*x**3 + 8*a*b**3*c*d*x**4 + 4*a*b**3*d**2*x**5 + b**4*c**2*x**4 + 2*b**4 *c*d*x**5 + b**4*d**2*x**6)),x)
(2*( - 256*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b*c**2*d**3 - 512*sqrt(d)*sq rt(b)*sqrt(a + b*x)*a**3*b*c*d**4*x - 256*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a* *3*b*d**5*x**2 - 768*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**2*c**2*d**3*x - 1536*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**2*c*d**4*x**2 - 768*sqrt(d)*sq rt(b)*sqrt(a + b*x)*a**2*b**2*d**5*x**3 - 768*sqrt(d)*sqrt(b)*sqrt(a + b*x )*a*b**3*c**2*d**3*x**2 - 1536*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b**3*c*d**4 *x**3 - 768*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b**3*d**5*x**4 - 256*sqrt(d)*s qrt(b)*sqrt(a + b*x)*b**4*c**2*d**3*x**3 - 512*sqrt(d)*sqrt(b)*sqrt(a + b* x)*b**4*c*d**4*x**4 - 256*sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**4*d**5*x**5 - 7 *sqrt(c + d*x)*a**5*d**5 + 105*sqrt(c + d*x)*a**4*b*c*d**4 + 70*sqrt(c + d *x)*a**4*b*d**5*x + 210*sqrt(c + d*x)*a**3*b**2*c**2*d**3 + 840*sqrt(c + d *x)*a**3*b**2*c*d**4*x + 560*sqrt(c + d*x)*a**3*b**2*d**5*x**2 - 70*sqrt(c + d*x)*a**2*b**3*c**3*d**2 + 420*sqrt(c + d*x)*a**2*b**3*c**2*d**3*x + 16 80*sqrt(c + d*x)*a**2*b**3*c*d**4*x**2 + 1120*sqrt(c + d*x)*a**2*b**3*d**5 *x**3 + 21*sqrt(c + d*x)*a*b**4*c**4*d - 56*sqrt(c + d*x)*a*b**4*c**3*d**2 *x + 336*sqrt(c + d*x)*a*b**4*c**2*d**3*x**2 + 1344*sqrt(c + d*x)*a*b**4*c *d**4*x**3 + 896*sqrt(c + d*x)*a*b**4*d**5*x**4 - 3*sqrt(c + d*x)*b**5*c** 5 + 6*sqrt(c + d*x)*b**5*c**4*d*x - 16*sqrt(c + d*x)*b**5*c**3*d**2*x**2 + 96*sqrt(c + d*x)*b**5*c**2*d**3*x**3 + 384*sqrt(c + d*x)*b**5*c*d**4*x**4 + 256*sqrt(c + d*x)*b**5*d**5*x**5))/(21*sqrt(a + b*x)*(a**9*c**2*d**6...